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A classic result in antiderivative theory

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November 28, 2019
functioncalculusreal analysisprimitives

Problem Statement

Let be a function f:R⟶R≥0 f:\mathbb{R}\longrightarrow\mathbb{R}_{\ge 0} that admits primitives and such that lim⁡x→0f(x)x=0. \lim_{x\to 0 } \frac{f(x)}{x} =0. Prove that the function g:R⟶R, g:\mathbb{R}\longrightarrow\mathbb{R} , defined as g(x)={f(x)/x,emsp;x≠00,emsp;x=0, g(x)=\left\{ \begin{matrix} f(x)/x ,&  x\neq 0\\ 0,&   x=0 \end{matrix} \right. , is primitivable.
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